From “what is a function?” to differential equations. Nothing assumed.
Calculus is the mathematics of things that change, and it is the first genuinely new idea most students meet after arithmetic and algebra. It is also the point at which a great many capable students quietly stop understanding and start following procedures.
This programme starts before the beginning — function notation, domain, what a limit actually is — and runs to differential equations. Every rule is derived rather than issued. If you have met calculus and it did not land, this is built for you as much as for someone starting cold.
What you need before starting: Light prior knowledge, not concept-zero. Every programme is built to be entered without prior exposure to the specific subject.
Every rule is derived rather than issued. The programme is designed to be entered without prior exposure to the subject — light prior knowledge, not concept-zero.
TopicsFunction notation and evaluation, domain/range, transformations, limit notation and interpretation
TopicsLimit laws, one-sided limits, continuity and discontinuity, IVT, algebraic and graphical limit methods
TopicsAverage vs instantaneous rate of change, derivative as a limit, derivative notation, tangent line slopes
TopicsPower rule, constant rule, sum and difference rules, product rule, quotient rule
TopicsChain rule, implicit differentiation, derivatives of exponential, logarithmic and trig functions
TopicsLogarithmic differentiation, inverse trig derivatives, hyperbolic functions, higher-order derivatives
TopicsVelocity and acceleration, related rates, position-velocity-acceleration relationships
TopicsCritical points, first and second derivative tests, absolute and relative extrema
TopicsIncreasing/decreasing intervals, concavity, inflection points, asymptotes, curve sketching
TopicsReal-world optimisation, business and economics applications, constrained optimisation
TopicsIndeterminate forms, L'Hôpital's Rule, tangent line approximations, differentials, Mean Value Theorem
TopicsComprehensive derivative review, problem-solving strategies, mixed applications, preparation for integration
TopicsAntiderivatives concept, indefinite integral notation, power rule for integration, constant of integration
TopicsU-substitution method, choosing substitutions, recognising patterns, reverse chain rule
TopicsIBP formula, choosing u and dv, repeated IBP, definite integrals by parts, DI method
TopicsIntegrals of trig functions, trig identities, powers and products of trig functions, reduction formulas
TopicsPartial fraction decomposition, linear factors, repeated factors, irreducible quadratic factors
TopicsTrigonometric substitution, improper integrals, convergence with infinite limits
TopicsRiemann sums, FTC Part 1 and Part 2, computing definite integrals
TopicsLinearity, additivity, comparison properties, average value, MVT for integrals
TopicsArea between curves, disk method, washer method, shell method for volume
TopicsArc length, surface area of revolution, work and force, fluid pressure, centre of mass
TopicsConsumer and producer surplus, present value, growth and accumulation models
TopicsComprehensive review, technique selection, multi-step problems, connecting differentiation and integration
TopicsDE definitions and order, general solutions, initial value problems, separating variables
TopicsExponential growth/decay, half-life, Newton's Law of Cooling, population dynamics
TopicsStandard form, integrating factor method, direction fields, logistic growth and equilibrium
TopicsEuler's method, accuracy and error, qualitative analysis, modelling with DEs
TopicsHomogeneous linear equations, characteristic equation, distinct real roots, general solution form
TopicsRepeated roots, complex roots and oscillations, particular solutions, damped oscillators, spring and circuit applications
Every session includes a problem set with full written solutions. Progress tracking and individual feedback throughout, and a conversational programme textbook written for the course.
A group opens when four students are ready.